Workflow management WvdA (2002)

1Wil van der Aalst.

Workflow managementWvdA (2002)

1

An Introduction to Petri-Nets

2

Basic concepts (static)

• State

– e.g. state of an elevator (5,-1)– snapshot of the process/system

• State space

– set of all possible states

6

5

4

3

2

1

3

Basic concepts (dynamic)

• A process/system has a behaviour, i.e. it moves from state tostate

time

stat

e

4

Discrete dynamic systems

• We focus on discrete processes, i.e. state changes are caused by events

time

stat

e

= event

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Example: elevator

• Graphical representation of atransition system

(Abstraction from time!)

-1 0 1

6

5

4

3

2

1

Floor

Direction6

A possible path

-1 0 1

6

5

4

3

2

1

7

Example: elevator

• A possible path representedby a state/time diagram

time

stat

e

(6,0)...

(3,1)(3,0)(3,-1)(2,1)(2,0)(2,-1)(1,1)(1,0)

-1 0 1

6

5

4

3

2

1

8

Other Examples

red

green

yellow

spring winter

summer autumn

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Other Examples (2)

on

off

error

start_course

pass

follow_lecture

exercise

do_exam

drop_out

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• Classical Petri-Nets• Modeling with Petri-Nets• Reachability Graphs• High-Level Petri-Nets

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Lecture Objectives

In this class, you are going to learn:• Classical Petri-Nets• Modeling with Petri-Nets• Reachability Graphs• High-Level Petri-Nets

12

High-level Petri nets

• The classical Petri net was invented by Carl Adam Petri in1962.

• A lot of research has been conducted (>10,000 publications).• Until 1985 it was mainly used by theoreticians.• Since the 80-ties the practical use is increasing because of

the introduction of high-level Petri nets and the availabilityof many tools.

• High-level Petri nets are Petri nets extended with– color (for the modeling of attributes)– time (for performance analysis)– hierarchy (for the structuring of models, DFD's)

4Wil van der Aalst.


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The classical Petri net model

A Petri net is a network composed of places ( ) and transitions ( ).

t2

p1

p2

p3

p4t3

t1

Connections are directed and between a place and a transition.Tokens ( ) are the dynamic objects.The state of a Petri net is determined by the distribution of tokens

over the places.

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Elements

(name)

(name)

place

transition

arc (directed connection)

token

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

place

transition

token

15

Rules

• Connections are directed.• No connections between two places or two transitions.• Places may hold zero or more tokens.

wait enter before make_picture after leave gone

free

occupied

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Transition t1 has three input places (p1, p2 and p3) and twooutput places (p3 and p4).

Place p3 is both an input and an output place of t1.

p1

p2

p3

p4t1

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Enabling condition for a Transition

Transitions are the active components and places and tokens arepassive.

A transition is enabled if each of the input places contains tokens.

t1 t2

Transition t1 is not enabled, transition t2 is enabled.

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Enabling a Transition


free

occupied

enabled Notenabled

Notenabled

19

Firing

An enabled transition may fire.Firing corresponds to consuming tokens from the input places and

producing tokens for the output places.

t2t2

Firing is atomic.

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Firing


free

occupied

fired

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Play “Token Game”

• In the new state, make_picture is enabled. It will fire, etc.


free

occupied

22

Firing: Example

23

Non-determinism

Two transitions fight for the same token: conflict.Even if there are two tokens, there is still a conflict.

t1

t2

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Non-determinism

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

transition t23

fires

t34 t43

t23 t32

t12 t21

t01 t10

p4

p3

p2

p1

p0

Two transitions areenabled but onlyone can fire

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Remarks

• Firing is atomic.• Multiple transitions may be enabled, but only one fires at a time• The number of tokens may vary if there are transitions for which the number of

input places is not equal to the number of output places.• The network is static.• The state is represented by the distribution of tokens over places (also referred

to as marking).

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Lecture Objectives


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Modeling with Petri Nets

• Place: passive element• Transition: active element• Arc: causal relation• Token: elements subject to change

The state (space) of a process/system is modeledby places and tokens.

State transitions are modeled by transitions (cf.transition systems).

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Role of a token

Tokens can play the following roles:• a physical object, for example a product, a part, a drug,

a person;• an information object, for example a message, a

signal, a report;• a collection of objects, for example a truck with

products, a warehouse with parts, or an address file;• an indicator of a state, for example the indicator of the

state in which a process is, or the state of an object;• an indicator of a condition: the presence of a token

indicates whether a certain condition is fulfilled.

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Role of a place

• a type of communication medium, like a telephoneline, a middleman, or a communication network;

• a buffer: for example, a depot, a queue or a post bin;• a geographical location, like a place in a warehouse,

office or hospital;• a possible state or state condition: for example, the

floor where an elevator is, or the condition that aspecialist is available.

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Role of a transition

• an event: for example, starting an operation, the deathof a patient, a change seasons or the switching of atraffic light from red to green;

• a transformation of an object, like adapting a product,updating a database, or updating a document;

• a transport of an object: for example, transportinggoods, or sending a file.

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Example: Single traffic lightrg

gy

yr

red

green

yellow

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Two traffic lights

rg

gy

yr

red

green

yellow

rg

gy

yr

red

green

yellow

rg

gy

yr

red

green

yellow

OR

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Problem

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Solution

rg1

gy1

yr1

r1

g1

y1

rg2

gy2

yr2

r2

g2

y2

x

How to makethemalternate?

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Two safe and fair traffic lights

rg1

red1

yellow1

green1

yr1

gy1

rg2

red2

yellow2

green2

yr2

gy2

safe2

safe1

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Example: life-cycle of a person

bachelor

child

married

puberty

marriage

divorce

death dead

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Lecture Objectives


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Multiple arcs connecting two nodes

• The number of arcs between an input place and a transitiondetermines the number of tokens required to be enabled.

• The number of arcs determines the number of tokens to beconsumed/produced.


free

39

Example: Ball game

• The number of arcs between two objects specifies the number oftokens to be produced/consumed.

• This can be used to model (dis)assembly processes.

blackred

bbrr

br

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• current stateThe configuration of tokens over the places.

• reachable stateA state reachable form the current state by firing a sequence ofenabled transitions.

• dead stateA state where no transition is enabled.

Some definitions

blackred

bbrr

br



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• 7 reachable states, 1 dead state.

blackred

bbrr

br (3,2)

(1,3) (3,1)

(1,2) (3,0)

(1,1)

(1,0)

rr

rr

rr

br

br

bb\br

bb\br

bb\br

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Exercise: your life-cycle

• How many states are reachable?• Is there a dead state?

dead

sleeping

active

start stop

die

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Exercise: readers and writers

• How many states are reachable?• Are there any dead states?• How to model the situation with 2 writers and 3 readers?• How to model a "bounded mailbox" (buffer size =4)?

rest

mail_box

receive_mail

type_mail

ready

rest

begin

send_mail

read_mail

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Lecture Objectives

In this class, we are going to learn:• Classical Petri-Nets• Modeling with Petri-Nets• Reachability Graphs• High-Level Petri-Nets



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Problems of Classical Petri nets

In practice the classical Petri net is not very useful:• The Petri net becomes too large and too complex, since

each case has its own set of attributes and needs one Petrinet for each set of attributes..

• It is not possible to handle time and data.• There is no support for structuring large models, e.g., top-

down and bottom-up design

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Models cannot reflect temporal aspects

rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

Duration of eachphase is highly

relevant.

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rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

No support for structuring large models

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High-level Petri nets

• To tackle the problems identified, Petri nets areextended with:– Color (i.e., data)– Time– Hierarchy



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To explain the three extensions we use the following exampleof a hairdresser's saloon.

start

waiting

finish

busy

free

ready

client waiting

hairdresser ready to begin

Note how easy it is to model the situation with multiple hairdressers.

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The extension with colorA token often represents an object having all kinds of attributes.Therefore, each token has a value (color) which refers to specific features

of the object modelled by the token.

start

waiting

finish

busy

free

ready

name: Harryage: 28experience: 2

name: Sallyage: 28hairtype: BL

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Extension with color

• Tokens have a color (i.e., a data value)

{Brand="BMW", RegistrationNo="GD-XW-11", Year=1993, Colour="blue", Owner= "Inge"}

{Brand="Lada", RegistrationNo="PH-14-PX", Year=1986, Color="grey", Owner="Inge"}

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Extension with color

• Places are typed (also referred to as color set).

{Brand="BMW", RegistrationNo="GD-XW-11", Year=1993, Colour="blue", Owner= "Inge"}

{Brand="Lada", RegistrationNo="PH-14-PX", Year=1986, Color="grey", Owner="Inge"}

record Brand:string

RegistrationNo:string

Year:int

Color:string

Owner:string



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Each transition has an (in)formal specification which specifies:• the number of tokens to be produced,• the values of these tokens,• and (optionally) a precondition.

The complexity is divided over the network and the values of tokens.

This results in a compact, manageable and natural process description.

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Examples

c := a+ba

b c

+

b := -a

b-a

if a> 0then b:= aelse c:=afi

a b

c

select

a >=0 | b := Ö a

bsqrta

Exercise:calculate Ö |a+b| using these buiding blocks

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Example: Making punch cards

start

donewait

stop

busy

free

waitingpatients

servedpatients

free deskemployees

patient/employees

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Tokens are colored

start

donewait

stop

busy

free

{Name="Klaas", Address="Plein 10", DateOfBirth="13-Dec-1962", Gender="M"}

{EmpNo=641112, Experience=7}



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Places are typed

start

donewait

stop

busy

free

record EmpNo:int * Experience:int

record Name:string *

Address:string *

DateOfBirth:str *

Gender:string

record Name:string *

Address:string *

DateOfBirth:str *

Gender:stringrecord Name:string *

Address:string *

DateOfBirth:str *

Gender:string *

EmpNo:int *

Experience:int58

Running example: Initial state

start

donewait

stop

busy

free



start is enabled

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Running example: Transition start fired

stop is enabled

start

donewait

stop

busy

free

{Name="Klaas", Address="Plein 10", DateOfBirth="13-Dec-1962", Gender="M", EmpNo=641112, Experience=7}

New value is created by simply merging the two records.

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Running example: Transition stop fired

New values are created by simply spliting the record into two parts.

start

donewait

stop

busy

free





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The number of tokens produced is no longer fixed (1)

test

samplenegative

positive

{sample_number=931101011, measurement_outcomes="XYV"}

{sample_number=931101023, measurement_outcomes="VXY"}

Note that the network structure is no longer a completespecification!

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The number of tokens produced is no longer fixed (2)

test

samplenegative

positive

{sample_number=931101011, measurement_outcomes="XYV"}

The number of tokens produced for each output place is between0 and 3 and the sum should be 3.

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The extension with timeFor performance analysis we need to model durations, delays, etc.Therefore, each token has a timestamp and transitions determine the delay

of a produced token.

start

waiting

finish

busy

free

ready

0

1

3 9

D=3

D=0D=0

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Extension with time

• Each token has a timestamp.• The timestamp specifies the earliest time when it can be

consumed.

2 5



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Extension with time

• The enabling time of a transition is the maximum of the tokens to be consumed.• If there are multiple tokens in a place, the earliest ones are consumed first.• A transition with the smallest firing time will fire first.• Transitions are eager, i.e., they fire as soon as they can.• Produced token may have a delay.• The timestamp of a produced token is the firing time plus its delay.

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Running example: Enabling time

• Transition start is enabled at time 2 = max{0,min{2,4,4}}.

start

donewait

stop

busy

free

4

2

4

0

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Running example: Delays

• Tokens for place busy get a delay of 3• @+3 = firing time plus 3 time units

start

donewait

stop

busy

free

4

2

4

0

@+3

@+0

@+0

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Running example: Transition start fired

• Transition start fired at time 2.

start

donewait

stop

busy

free

45

4

@+0

@+0

@+3

Continue to play (timed) token game…



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Exercise: Final state?

x

y

d

e

b

a

c

4

3

5

1

@+1

@+2

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Extension with hierarchy

• Timed and colored Petri nets result in more compact models.• However, for complex systems/processes the model does not fit on

a single page.• Moreover, putting things at the same level does not reflect the

structure of the process/system.• Many hierarchy concepts are possible. In this course we restrict

ourselves to transition refinement.

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rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

Instead of

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rg1

go1

or1

r1

g1

o1

rg2

go2

or2

r2

g2

o2

x

tl1 tl2

x

x

We can use hierarchy



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rg

go

or

r

g

tl1 tl2

x

x

o

Reuse

• Reuse saves design efforts.• Hierarchy can have any

number of levels• Transition refinement can

be used for top-down andbottom-up design

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The extension with hierarchy• A subnet is a net composed out of places, transitions and subnets.

waiting ready

h1

h2

h3

start finishbusy

free

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Exercise: remove hierarchy

waiting ready

h1

h2

h3

start finishbusy

free

begin endpending

begin endpending

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Conclusions

• Petri net is a tool that allows processes to be modeledgraphically and analyzed in a formal way.

• Petri net forms the basis of many process definitiontools available in the market.

• Classical Petri net needs to be extended with color, timeand hierarchy in order to make it capable of modelingbusiness processes.

• Next, we study how workflow processes and resourcemanagement can be modeled with Petri-net.

Documents

Workflow management WvdA (2002)